CAPM extensions in the literature

Extensions of the mean variance framework have a long history in the academic literature. Arditti and Levy (1972) show that non-increasing absolute risk aversion implies investor preference for positive skewness, while Rubinstein (1973) provides a model in which expected returns are equal to the weighted sum of higher co-moments. Horvath (1980) shows that risk averse investors with decreasing marginal utility have a positive preference for mean and skewness and a negative preference for variance and kurtosis, and thus risk-averse investors prefer higher returns and skewness, and lower variance and kurtosis.

Kraus and Litzenberger (1976) derive a three-moment CAPM by adding coskewness risk to the standard CAPM and apply it for portfolios of stocks double-sorted on beta and systematic coskewness over the period 1936-1970 using the Fama and MacBeth (1973) methodology. They find an insignificant negative intercept, a significant positive beta premium (larger than that obtained in a model when beta is the only explanatory), and a market premium for gamma which is significant and negative, consistent with expectations. However, Friend and Westerfield (1980) find results that partly contradict the findings of Kraus and Litzenberger, with a significant non-zero intercept and a time-varying coefficient for co-skewness.

Lim (1989) tests the three-moment CAPM using a generalised method of moments (GMM) approach and shows that investors prefer coskewness when market returns are positively skewed, and dislike coskewness when market returns are negatively skewed. Harvey and Siddique (2000) test whether the inclusion of various measures of conditional co-skewness improves the pricing errors in the three-factor model of Fama and French and the standard CAPM. They show that the adjusted R-squared statistics of both models improve.

Fang and Lai (1997) test the four-moment CAPM on portfolios triple-sorted on beta, coskewness, and cokurtosis over three distinct five-year periods where the factor loadings are estimated using time series regressions of the cubic market model.¹ The results show a substantial improvement in R-squared for the four-moment CAPM compared to the two- and three-moment CAPM. Most importantly, the risk premia for beta and cokurtosis are significant and positive for the three sub-periods. Athayde and Flores (1997) test a four-moment CAPM for Brazilian stocks using GMM over the period 1996-1997. They conclude that skewness, rather than kurtosis, plays the most important role for the Brazilian stocks. Hwang and Satchell (1998) estimate an unconditional four-moment CAPM for emerging market stocks over the period 1985-1997 using GMM and conclude that higher moments can add explanatory power to model returns for emerging markets, though with variations across countries.

Dittmar (2002) estimates a conditional four-moment CAPM using a stochastic discount factor approach. Two models are implemented: one with the equity market index as a market proxy of wealth and another including human labour wealth. The model terms are found significant and the pricing errors are significantly reduced when human capital is included in the four-moment CAPM. Tan (1991) applies a three-moment CAPM on a sample of mutual funds and finds results that do not support the three-moment CAPM (an intercept significantly greater than zero, an insignificant beta risk premium, and an incorrect positive sign for coskewness). Hasan and Kamil (2013) test a higher-moment CAPM with coskewness, cokurtosis, market capitalization and book-to-market to model stock returns in Bangladesh, and find that coskewness and cokurtosis are weakly negatively and positively related to returns, respectively.

Lambert and Hubner (2013) test an extension of the four-moment CAPM for US returns over the period 1989-2008. When testing an augmented three-factor model of Fama and French with coskewness and cokurtosis on the 25 portfolios sorted on size and book-to-market, they find that low book-to-market portfolios are positively related to high cokurtosis risk and small sized portfolios are positively related to low coskewness, that is, they are more exposed to coskewness risk. In the cross-section they find that the augmented model reduces the pricing errors and has a higher R-squared than the simple CAPM. Kostakis et al. (2011) test the higher-moment CAPM for UK stocks over the period 1986-2008 and find that coskewness demands a negative risk premium whereas stocks with higher cokurtosis yield higher returns on average. In particular, coskewness and cokurtosis have additional explanatory power to covariance risk, size, book-to-market and momentum factors. The alpha or unexplained return of portfolios with negative coskewness and positive cokurtosis is not eliminated in time series after controlling for size, value and the momentum factors of Carhart (1997).

Young et al. (2010) examine a higher-moment CAPM in which the moments are estimated using daily data for S&P500 index options. They find that stocks with high exposure to change in implied market volatility and market skewness yield lower returns whereas stocks with higher sensitivity to kurtosis yield higher returns. Heaney et al. (2012) test whether coskewness and cokurtosis are priced for US equity returns over the period 1963-2010, using individual assets as opposed to portfolios. They find little evidence that the higher moments are priced and show that these are encompassed by size and book-to-market factors. In particular, size tends to eliminate the significance of cokurtosis, which is found unexpectedly to be negatively rewarded, and coskewness varies over time. Furthermore, Moreno and Rodriguez (2009) analyse the returns of US mutual funds over the period 1962-2006 and find that coskewness is a priced risk factor for mutual funds. The findings show that funds that invest in stocks with more negative coskewness tend to yield higher average returns. Finally, Doan et al. (2008) analyse a higher-moment CAPM for US and Australian stocks and find that returns are sensitive to higher moments, and that higher moments explain a portion of returns not explained by the Fama and French factors. Thus, there is now a significant body of literature on the higher moment CAPM which underpins the importance of considering skewness and kurtosis when modelling asset returns, though the results of empirical studies do not as yet offer conclusive support for the model.

We use the monthly returns CRSP data for all common stocks listed on the NYSE, AMEX and NASDAQ exchanges for the period January 1926 to December 2010 (NASDAQ from January 1972). The empirical tests are conducted for the sample 1930-2010, and for the subsample 1980-2010. In the cross-sectional regressions used in the empirical tests, only stocks with 24 months of returns are included, as this is required for our short window beta computation.

Table 1 reports summary statistics for the cross section of stock returns at 10 year intervals over the sample period. The number of stocks increased over the years from less than 1,000 stocks before 1950 to a peak of 9,681 in December 2000. The number of stocks appears to be driven by the trend of industrialization and the economic boom period from the 1980 to 2000. We also observe that the standard deviation of the returns across stocks has increased since the 1980s, thus suggesting two distinct volatility regimes: before and after 1980. The normality of the distribution of returns, that is the joint hypothesis that skewness and excess kurtosis are equal to zero, is rejected in the cross-section for each period in the table.
[Insert Table 1 here]

The testing of the CAPM is straightforward. Re-writing Equation 1 as

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For the market portfolio:

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The CAPM and the four-moment CAPM are typically tested on portfolios. However, many authors (see, for instance, Ang et al., 2008; and Kim, 1995) have criticized the use of portfolios to estimate the market premium, arguing that the spread in betas is effectively too small when portfolios are formed, leading to very large standard errors in the estimation of the risk premium. Further, Kim (1995) argues that when portfolios are formed, the behaviour of individual stocks is smoothed out, losing important information for the estimation of the risk premium in the process. We therefore use individual assets in our tests.

We address the limitations of static models by adopting the approach of Lewellen and Nagel (2006). We thus employ a two-step method. First, we conduct short-window (24 month) time series regressions of monthly individual asset excess returns over the market excess return to estimate conditional betas, coskewness, and cokurtosis as in Kraus and Litzenberger (1976). The average excess return for the individual stocks over the short windows is assumed to be their (conditional) expected excess return. In the second step, the average excess returns of individual stocks are regressed in the cross section over the conditional co-moments (calculated over the same window as in the first step) to estimate the risk premia. The monthly conditional risk premia are then treated as time series observations, and hence tested using the Fama and MacBeth (1973) approach, i.e. the monthly conditional risk premia for each factor and the overall monthly conditional risk premia are averaged. However, because of potential autocorrelation and heteroscedasticity problems in the monthly risk premia, we also show results based on intercept only GMM estimation with Newey-West heteroscedasticity and autocorrelation consistent (HAC) standard errors. The models are conditional in so far as the factor loadings are obtained using rolling short windows.

The Fama and French factor augmented four-moment CAPM

The four-moment CAPM can be extended to include the effect of the Fama and French (1993) SMB and HML factors. The SMB factor represents the return of a portfolio of small-capitalization stocks minus the return on a portfolio of large-capitalization stocks. The HML factor represents the returns of a portfolio of high book-to-market stocks minus the return on a portfolio of low book-to-market stocks. The three-factor model of Fama and French is widely used in empirical studies of asset pricing. Therefore it would be interesting to augment the conditional four-moment CAPM with the SMB and HML factors in order to determine whether the Fama-French factors remain relevant in an asset pricing model once we correct for additional moment sensitivities.

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